Sunday, October 14, 2007

Propositional Logic with Literal Diagrams

I had been hoping to have a post on this much earlier (I think it's quite cool); but I have been a combination of busy, tired, and not feeling well for several days now. Having a bit of breathing space, I can finally note it. Propositional logic is often taught as if the only ways to test groups of statement in propositional logic for consistency were truth tables and truth trees. But we can also easily handle simple statements with Carroll's literal diagrams, in a way that is far more clear than either truth tables or truth trees.

A literal diagram is a diagram that recognizes combinations of terms. The standard biliteral diagram looks something like this, with the boxes given labels:

 +p+q +p-q -p+q -p-q

In propositional logic this is understood to represent possible ways the whole world can be, given the truth (+) or falsehood (-) of p and q. Information about p and q 'black out' possible ways the world could be given that information. With this insight we can easily represent all the propositional connectives:

 x x x

This is the biliteral diagram for p & q. The biliteral diagram for p v q is:

 x

The biconditional p ↔ q is:

 x x

The conditional, p → q is:

 x

Now, take the following pair of sentences: p ↔ q, p, -q. The result is a world blackout, i.e., the conclusion that there is no possible way for the world to be given these three statements together:

 x x x x

p blacks out the -p+q and -p-q; -q would black out out +p+q (and -p+q, but p already blacks it out). This only leaves +p-q. But this (along with the redundant -p+q again) is blacked out by p ↔ q. World blackout: the statements taken together are inconsistent.

Another example. The statements (p v -q), (-p), and (-p & q) lead to world blackout as well. -p blacks out +p+q and +p-q. -p&q blacks out all except -p+q. But -p+q is precisely what (p v -q) blacks out.

Diagrams with world blackouts always indicate inconsistent statements; diagrams with at least one space open are consistent. And if we are dealing with 3 propositions we can do the same thing with a triliteral diagram, and so forth.

It's also easy to do two other useful things with literal diagrams. First, we can easily tell whether two statements are equivalent, by seeing whether they are diagrammed the same way. Second we can easily tell that statement B is not implied by statement A by diagramming them and determining whether B is part of A. If B, for instance, has information not contained in A, then it is not implied by it. If, on the other hand, B's diagram is nothing more than part of A's diagram, A implies B. (In effect doing these comparisons is like making a higher-order diagramming showing A ↔ B or A → B, whichever is being considered, and whatever A and B might be.)

Of course, it is no mystery why literal diagramming works in this case; the literal diagram, interpreted for propositional logic, is logically equivalent to a truth table. We can think of them as truth diagrams. The advantage to working with them is that they make explicit and visual the concepts of inconsistency, equivalence, and implication. Their only disadvantage in comparison with standard truth tables, actually, is that they quickly become unwieldy as you multiply the number of propositions you work with. But one can easily imagine beginning with truth diagrams, and using them to introduce truth tables.

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